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 Post subject: Unusual killer variant
PostPosted: Tue Dec 04, 2012 5:37 pm 
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For those of you who don't frequent The New Sudoku Players' forum, I ran across a puzzle in a thread there that may interest you. The OP in that thread was asking about software to help solve it. I've never before seen a killer that wasn't 9x9 (or a multi-grid variant thereof). Here's the thread:

http://forum.enjoysudoku.com/post222510.html#p222510

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PostPosted: Tue Dec 04, 2012 10:38 pm 
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Thanks enxio27 for posting this 12x12 killer. It's not quite unique in not being a 9x9 killer. Here are some others which I've come across.

Mike (mhparker) posted a pair of 8x8 killers Cyclone V1 and V2 in the early days of this forum. The solutions are clearer in the archive entries Cyclone V1 and Cyclone V2.

More recently HATMAN's Paper Solvable 6 Decidoku X Killer was a 10x10 killer, but this used candidates 0-9.

I'll try this 12x12 killer, so I hope it has a unique solution. I'll have to forget some combinations which are "ingrained" in my memory; a 17(2) cage for this puzzle isn't necessarily {89}. I'll probably also need to use a calculator; I don't trust my mental arithmetic to calculate multiples of 78 (the total for each row, column and box).


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PostPosted: Fri Dec 07, 2012 1:47 am 
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An enjoyable puzzle; this 12x12 killer-X proved to be easier than expected. No need to use a calculator; I just used the cursor on my Excel worksheet to get sum cage totals, then mental arithmetic to get the difference from 78 or 156 (2x78) - I don't think I'm giving away anything by saying that I didn't need to use higher multiples of 78.

Hint:
Don't get put off by thinking that combinations of 12 different numbers are difficult. Use the 78 rule as much as you can from the start. It will give you some immediate placements and some useful 2-cell hidden cages.

For ease of description I used the letters A, B and C to represent candidate values 10, 11 and 12 in my walkthrough; I hope that doesn't confuse anyone who looks at it. The solution is given as two diagrams, one with 1-9, A-C and the other with 1-12.

Here is my walkthrough for Unusual Killer Variant:
This is a 12x12 Killer-X. Not sure what the correct term is for the 12-cell equivalent of a nonet, so I’ll just call it a Box, numbering B1 to B12. For simplicity I’m calling the candidates 1-9, then A=10, B=11 and C=12, rather than 1-12, since it makes diagrams easier.

For a 12x12 killer, the 45 rule becomes the 78 rule.

I’ll skip Prelims; even for 2-cell cages, most won’t eliminate many candidates.

1. 78 rule on B1 1 innie R3C1 = 7

2. 78 rule on B12 1 innie R10C12 = 7

3. 78 rule on B2+B3 1 outie R4C8 = 3, R3C8 = 8

4. 78 rule on B10+B11 1 outie R9C5 = 7, R10C5 = 3
4a. 9(2) cage at R1C5 = {45} (only remaining combination), locked for C5 and B2

5. 21(2) cage at R1C8 = {9C/AB}
5a. 21(2) cage at R1C10 = {9C/AB}
5b. Naked quad {9ABC} in 21(2) cages, locked for R1

6. 13(2) cage at R1C6 = {67} (only remaining combination), locked for R1 and B2

7. 78 rule on R1 3 innies R1C1 + R1C5 + R1C12 = 9 = {135/234}, 3 locked for R1
7a. R1C5 = {45} -> R1C1 + R1C12 = {13/23}
7b. 14(3) cage at R1C2 = {158/248}, 8 locked for B1

8. 78 rule on B3 2 outies R1C8 + R2C8 = 19 = {9A} (only remaining combination), locked for C8 and B2, clean-up: no 9,A in R1C9
8a. B,C in R1 only in R1C9 + R1C10 + R1C11, locked for B3

9. 12(2) cage at R3C5 = {1B} (only remaining combination), locked for R3 and B2

10. 20(2) cage at R5C7 = [8C/9B] -> R5C7 = {89}, R5C8 = {BC}

11. 78 rule on B10 2 outies R11C5 + R12C5 = 20 = {8C/9B}
11a. 15(2) cage at R12C4 = [3C/4B/69/78], R12C3 = {3467}
11b. Min R11C5 = 8 -> max R11C34 = 7, R11C34 = {123456}

12. 18(2) cage at R8C4 = {6C/7B/8A}, no B in R8C4

13. 78 rule on B7 2 innies R7C34 = 19 = {7C/8B/9A}
13a. 16(2) cage at R6C3 = [4C/5B/6A/79/97], no 8 in R7C3, no B in R7C4

14. 78 rule on B4 2 remaining innies R4C4 + R6C3 = 16 = [79/97/A6/B5/C4]
14a. 17(2) cage at R4C4 = [7A/98/B6], clean-up: no 4,6 in R6C3, no A,C in R7C3, no 7,9 in R7C4 (step 13)

15. 78 rule on B8 1 outie R7C4 = 1 remaining innie R9C8 + 7 -> R7C4 = {8C}, R9C8 = {15}, clean-up: no 9 in R7C3 (step 13), no 7 in R6C4, no 9 in R4C4 (step 14), no 8 in R4C5
15a. 11(2) cage at R9C8 = [1A/56]
15b. 18(2) cage at R8C4 (step 12) = {6C/8A} (cannot be {7B} which clashes with R7C3), no 7,B
15c. Killer pair 8,C in R7C4 and 18(2) cage, locked for C4 and B7

16. 6(2) cage at R7C8 = {24} (only remaining combination, cannot be {15} which clashes with R9C8), locked for C8 and B8

17. 21(2) cage at R7C7 = {9C/AB }

18. 13(2) cage at R11C8 = {1C/67}

19. 14(2) cage at R10C7 = [2C/86/95]

20. 13(2) cage at R9C6 = {1C/3A/58}

21. 13(2) cage at R9C6 (step 20) = {1C/3A/58}, 11(2) cage at R9C8 (step 15a) = [1A/56] -> combined cage R9C6789 = {1C}[56]/{3A}/[56]/{58}[1A], 5 locked for R9 and B8

22. 15(2) cage at R8C5 = [69/96/C3]
22a. 21(2) cage at R7C7 (step 17) = {AB} (only remaining combination, cannot be 9C which clashes with 15(2) cage), locked for C7 and B8

23. 13(2) cage at R9C6 (step 20) = {1C/58}, no 3
23a. Killer pair 1,5 in 13(2) cage and R9C8, locked for R9 and B8

24. 23(3) cage at R7C4 = [869/896/8C3/C83], R7C6 = {369}, 8 locked for R7
24a. 3 in B8 only in R78C6, locked for C6

25. 14(2) cage at R5C5 = {2C/68}

26. R4C5 = A (hidden single in C5), R4C4 = 7, placed for D\, R6C3 = 9 (step 14), R7C3 = 7, R7C4 = C (step 13)
26a. 18(2) cage at R8C4 (step 15b) = {8A} (only remaining combination), locked for C4 and B7

27. 23(3) cage at R7C4 (step 24) = [C83] (only remaining permutation) -> R7C5 = 8, R7C6 = 3, placed for D/

28. R1C1 = 3 (hidden single in R1), placed for D\

29. 14(2) cage at R5C5 (step 25) = {2C} (only remaining combination), locked for C5 and B5 -> R5C8 = B, placed for D/, R5C7 = 9, clean-up: no 5 in R10C8
29a. Naked pair {9B} in R11C5 + R12C5, locked for C5 and B11 -> R3C5 = 1, R3C6 = B, R8C5 = 6, placed for D/, R8C6 = 9

30. 10(2) cage at R4C6 = {46} (only remaining combination), locked for R4 and B5

31. 9(2) cage at R5C6 = {18} (only remaining combination), locked for C6 and B5

32. Naked pair {57} in R6C78, locked for R6, R6C9 = A (cage total), R9C9 = 6, placed for D\, R9C8 = 5, R6C8 = 7, R6C7 = 5, placed for D/, R9C6 = C, R9C7 = 1, R2C6 = 2

33. 13(2) cage at R11C8 (step 18) = {1C} (only remaining combination), locked for B11 -> R10C8 = 6, R10C7 = 8

34. 8(2) cage at R7C9 = [17/53]
34a. 8(2) cage at R7C11 = [17/53]
34b. Naked pair {15} in R7C9 + R7C11, locked for R7 and B9
34c. Naked pair {37} in R8C9 + R8C11, locked for R8 and B9

35. 78 rule on B9+B12 1 outie R6C10 = 6, R7C10 = B, R78C7 = [AB]

36. 24(3) cage at R8C10 = {2AC/48A}, no 9, A locked for B9
36a. 9 in B9 only in R79C12, locked for C12

37. 15(2) cage at R12C4 = [4B/69], no 3
37a. R56C4 = [24/42/51]

38. R11C4 = 3 (hidden single in C4)
38a. 78 rule on B10 2 remaining innies R11C3 + R12C4 = 7 -> R11C3 = 1, R12C4 = 6, R12C5 = 9, R11C5 = B, R11C8 = C, R12C8 = 1

38. 78 rule on R12 2 innies R12C1 + R12C12 = 13 = [2B/85]
38a. 9(2) cage at R12C6 = [54/72]
38b. Killer pair 2,5 in R12C1 + R12C12 and 9(2) cage, locked for R12

39. 3 in R12 only in 25(3) cage at R12C9 = {3AC} (only remaining combination), locked for R12 and B12

40. A in C12 only in 14(3) cage at R1C12 = {13A} (only remaining combination) -> R1C12 = 1, placed for D/, R23C12 = {3A}, locked for C12 and B3

41. 21(2) cage at R1C10 = {9C} (only remaining combination), locked for R1 and B3 -> R1C89 = [AB]

42. R2C8 = 9 -> 22(3) cage at R2C8 = {589} (only remaining combination), locked for R2, 5,8 also locked for B3 -> R2C5 = 4, R1C5 = 5

43. Naked pair {24} in R3C9 + R3C10, locked for R3 and B3 -> R2C11 = 7, placed for D/, R3C11 = 6, R8C11 = 3, R7C11 = 5, R78C9 = [17]

44. 6(2) cage at R5C4 = [51] (cannot be {24} which clashes with R1C4), R6C6 = 8, placed for D\, R5C6 = 1

45. R23C4 = [B9] = 20 -> R23C3 = 11 = {56} -> R2C3 = 6, R3C3 = 5, placed for D\, R12C12 = B, R12C1 = 2 (step 38), placed for D/, R3C10 = 4, placed for D/, R3C9 = 2, R12C7 = 4, R12C6 = 5, R4C67 = [46], R1C67 = [67], R10C6 = A, R11C67 = [72], R12C23 = [78]

46. R10C4 = 4, R10C3 = C, placed for D/
46a. R10C34 = [C4] = 16 -> R10C2 + R11C2 = 19 = {9A} -> R10C2 = 9, R11C2 = A, placed for D/, R9C4 = 8, placed for D/, R8C4 = A, R4C9 = 9, R5C9 = 3

47. 14(3) cage at R1C2 = [842], R8C3 = 2, R8C8 = 4, placed for D\, R7C8 = 2, R11C11 = 9, R3C2 = C, R2C2 = 1, placed for D\, R2C1 = A, R10C9 = 5, R10C10 = 2, placed for D\, R5C5 = C, R6C5 = 2

48. 9(2) cage at R5C2 = {36} (only remaining combination) -> R5C2 = 6, R6C2 = 3

49. R23C7 = [C3], R2C9 = 8, R2C10 = 5, R23C12 = [3A], R7C2 = 4, R7C12 = 9, R7C1 = 6, R8C1 = 1, R8C2 = 5, R10C1 = B, R11C1 = 5, R10C11 = 1, R11C9 = 4, R11C10 = 8, R11C12 = 6, R8C10 = C, R1C10 = 9, R1C11 = C, R45C10 = [17], R9C10 = A, R9C11 = 2 (cage sum), R8C12 = 8, R9C12 = 4, R12C9 = C, R12C10 = 3, R12C11 = A, R5C12 = 2, R4C23 = [2B], R4C11 = 8, R4C1 = C, R4C12 = 5, R56C1 = [84], R5C3 = A, R56C11 = [4B], R6C12 = C, R9C123 = [9B3]

Solution:
Image    Image

Rating Comment:
I'll rate my walkthrough at Easy 1.25. I used one combined cage which was a useful shortcut.


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